, Volume 327, Issue 4, pp 723-744
Date: 30 Oct 2003

Blow-up solutions of nonlinear elliptic equations in ℝ n with critical exponent Dedicated to Philippe Dufour, whose exquisite horology artwork ‘‘Simplicity’’ inspires so much.

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Abstract

For an integer n≥3 and any positive number ɛ, we establish the existence of smooth functions K on ℝ n ∖{0} with |K−1|≤ɛ, such that the equation \({{ \Delta u + n (n - 2) K u^{{{{n + 2}}}\over{ {{n - 2}}}} = 0 {{{{\rm{ in}}}}} {{{{\mathbb R}}}}^n \setminus \{ 0 \} }}\) has a smooth positive solution which blows up at the origin (i.e., u does not have slow decay near the origin). Furthermore, we show that in some situations K can be extended as a Lipschitz function on ℝ n . These provide counter-examples to a conjecture of C.-S. Lin when n>4, and a question of Taliaferro.

Mathematics Subject Classification (2000) Primary 35J60; Secondary 53C21